arXiv:cs/0508069v2 [cs.LO] 22 Feb 2006

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Real Hypercomputation and Continuity⋆

Martin Ziegler⋆⋆

University of Paderborn, 33095 GERMANY

Abstract. By the sometimes so-calledMain Theorem of Recursive Anal-ysis, every computable real function is necessarily continuous. We won-der whether and which kinds of hypercomputation allow for the effectiveevaluation of also discontinuous f : R → R. More precisely the presentwork considers the following three super-Turing notions of real functioncomputability:

– relativized computation; specifically given oracle access to theHalting Problem ∅′ or its jump ∅′′;

– encoding input x ∈ R and/or output y = f(x) in weaker ways alsorelated to the Arithmetic Hierarchy;

– non-deterministic computation.

It turns out that any f : R → R computable in the first or second sense isstill necessarily continuous whereas the third type of hypercomputationdoes provide the required power to evaluate for instance the discontinu-ous Heaviside function.

1 Motivation

What does it mean for a Turing Machine, capable of operating only on discreteobjects, to compute a real number x:

ρb,2ρb,2ρb,2: To determine its binary expansion, i.e., to decide A ⊆ N with x =∑

n∈A

2−n ?

ρCnρCnρCn: To compute a sequence (qn) of rational numbers eventually converging to x?

ρρρ: To compute a fast convergent sequence (qn) ⊆ Q for x, i.e. with |x− qn|≤ 2−n (in other words: to approximate x with effective error bounds)?

ρ<ρ<ρ<: To approximate x from below, i.e., to compute (qn) such that x = supn qn ?

All these notions make sense in being closed under arithmetic operations likeaddition and multiplication. In fact they are well (known equivalent to variants)studied in literature⋆ ⋆ ⋆; e.g. [Tur36], [BH02], [Tur37], [Wei01] in order.

Now what does it mean for a Turing Machine M to compute a real func-tion f : R → R? Most naturally it means that M realizes effective evaluationx 7→ f(x) in that, upon input of x ∈ R given in one of the above ways, itoutputs y = f(x) also in one (not necessarily the same) of the above ways.

⋆ An extended abstract of this work, mostly lacking proofs, has appeared as [Zie05].⋆⋆ supported by the German Research Fundation DFG with project Zi1009/1-1

⋆ ⋆ ⋆ Their above names by indexed Greek letters are taken from [Wei01, Section 4.1].

2 Martin Ziegler

And, again, many possible combinations have already been investigated. For in-stance the standard notion of real function computation in Recursive Analysis[Grz57,PER89,Ko91,Wei01] refers (or is equivalent) to input and output givenaccording to ρ. Here, the Main Theorem of Computable Analysis implies thatany computable f will necessarily be continuous [Wei01, Theorem 4.3.1].

We are interested in ways of lifting this restriction, that is, in the following

Question 1. Does hypercomputation in some sense permit the computationalevaluation of (at least certain) discontinuous real functions?

That is related to the Church-Turing Hypothesis: A Turing Machine’s abilityto simulate every physical process would imply all such processes to behavecontinuously—a propertyG. Leibniz was convinced of (“Natura non facit saltus”)but which we nowadays know to be violated for instance by the Quantum HallEffect awarded a Nobel Prize in 1985. Since this (nor any other) discontinuousphysical process cannot be simulated on a classical Turing Machine, it constitutesa putative candidate for a system capable of realizing hypercomputation.

1.1 Summary

The standard (and indeed the most general) way of turning a Turing Machineinto a hypercomputer is to grant it access to an oracle like, say, the Halting Prob-lem ∅′ or its iterated jumps like ∅′′ and ∅(d) in Kleene’s Arithmetic Hierarchy.However regarding computational evaluation of real functions, closer inspectionin Section 3.1 reveals that this Main Theorem relies solely on information ratherthan recursion theoretic arguments and therefore requires continuity also fororacle-computable real functions with respect to input and output of form ρ.(For the special case of an ∅′–oracle, this had been observed in [Ho99, Theo-rem 16].)

A second idea, applicable to real but not to discrete computability, changesthe input and output representation for x and y = f(x) from ρ to a weakerform like, say, ρCn. This relates to the Arithmetic Hierarchy, too, however in adifferent way: Computing x in the sense of ρCn is equivalent to computing x inthe sense of ρ [Ho99, Theorem 9] relative (i.e., given access) to the HaltingProblem ∅′ and thus suggests to write ρ′ := ρCn. Most promisingly, the MainTheorem [Wei01, Corollary 3.2.12] which requires continuity of (ρ → ρ)–computable real functions applies to ρ but not to ρ′ because the latter lacks thetechnical property of admissibility.

It therefore came to quite a surprise when Brattka and Hertling estab-lished that any (ρ′ → ρ′)–computable f (that is, with respect to input x andoutput f(x) encoded according to ρCn) still satisfies continuity; see [Wei01, Ex-ercise 4.1.13d] or [BH02, Section 6].

Section 3.2 contains an extension of this and a series of related results.Specifically we manage to prove that continuity is necessary for (ρ′′ → ρ′′)–computability of f ; here, ρ ¬ ρ′ ¬ ρ′′ ¬ . . . denote the first levelsof an entire hierarchy of real number representations explained in Lemma 5

Real Hypercomputation and Continuity 3

which emerge naturally from the Real Arithmetic Hierarchy of Weihrauchand Zheng [ZW01].

In Section 4, we closer investigate the two approaches to real function hy-percomputation. Specifically it is established (Section 4.1) that the hierarchy ofreal number representation actually does yield a hierarchy of weakly computablereal functions. Furthermore a comparison of both oracle-supported and weaklycomputable (and each hence necessarily continuous) real functions in Section 4.2reveals a relativized version of the Effective Weierstraß Theorem to fail.

Our third approach to real hypercomputation (Section 5) finally allows theTuring Machines under consideration to behave nondeterministically. Remark-ably and in contrast to the classical (Type-1) theory, this does significantly in-crease their principal capabilities. For example, all quasi-strongly δ–Q–analyticfunctions in the sense of Chadzelek and Hotz [CH99]—and in particularmany discontinuous real functions—now become computable as well as conver-sion among the aforementioned representations ρCn and ρb,2.

2 Arithmetic Hierarchy and Reals

In [Ho99], Ho observed an interesting relation between computability of a realnumber x in the respective senses of ρ and ρCn in terms of oracles: x = limn qnfor an (eventually convergent) computable rational sequence (qn) iff x admits afast convergent rational sequence computable with oracle ∅′, that is, a sequence(pm) ⊆ Q recursive in ∅′ with |x − pm| ≤ 2−m. This suggests to use ρ′ synony-mously for ρCn; and denoting by ∆1R = Rc the set of reals computable in thesense of Recursive Analysis (that is with respect to ρ), it is therefore naturalto write, in analogy to Kleene’s classical Arithmetic Hierarchy, ∆2R for the setof all x ∈ R computable with respect to ρ′. Weihrauch and Zheng extendedthese considerations and obtained for instance [ZW01, Corollary 7.3] the fol-lowing characterization of ∆3R: A real x ∈ R admits a fast convergent rationalsequence recursive in ∅′′ iff x is computable in the sense of ρ′′ defined as follows:

ρ′′ρ′′ρ′′: x = limi limj q〈i,j〉 for some computable rational sequence (qn)

where 〈· · · 〉 : N∗ → N denotes some fixed computable pairing or, more generally,tupling function. Similarly, Σ1R contains of all x ∈ R computable with respect toρ< whereas Σ2R includes all x computable in the sense of ρ′< defined as follows:

ρ′<ρ′<ρ′<: x = supi infj q〈i,j〉 for some computable rational sequence (qn).

To Σ2R belongs for instance the radius of convergence r = 1/ lim supn→∞n√an

of a computable power series∑∞

n=0 anxn [ZW01, Theorem 6.2]. More generally

we take from [ZW01, Definition 7.1 and Corollary 7.3] the following

4 Martin Ziegler

Definition 2 (Real Arithmetic Hierarchy). Let d = 0, 1, 2, . . .

ρ(d)<ρ(d)<ρ(d)< : Σd+1R consists of all x ∈ R of the form x = sup

n1

infn2

. . . Θnd+1

q〈n1,...,nd+1〉

for a computable rational sequence (qn),where Θ=sup or Θ=inf depending on d’s parity;

ρ(d)>ρ(d)>ρ(d)> : Πd+1R similarly for x = inf

n1

supn2

. . .

ρ(d)ρ(d)ρ(d): ∆d+1R contains all x ∈ R of the form x = limn1

limn2

. . . limnd

q〈n1,...,nd〉

for a computable rational sequence (qn).

(For an extension to levels beyond ω see [Bar03]. . . )The close analogy between the discrete and this real variant of the Arithmetic

Hierarchy is expressed in [ZW01] by a variety of elegant results like, e.g.,

Fact 3. a) x ∈ ∆dR iff deciding its binary expansion is in ∆d.b) x is computable with respect to ρ(d)

iff there is a ∅(d)–computable fast convergent rational sequence for x.

c) x is computable with respect to ρ(d)<

iff x is the supremum of a ∅(d)–computable rational sequence.d) ∆dR = ΣdR ∩ ΠdR.e) ΣdR ∪ ΠdR ( ∆d+1R.

Proof. a) Theorem 7.8, b+c) Lemma 7.2, d) Definition 7.1, and e) Theo-rem 7.8 in [ZW01], respectively. ⊓⊔

2.1 Type-2 Theory of Effectivity

Specifying an encoding formalizes how to feed some general form of input likegraphs or integers into a Turing Machine with fixed alphabet Σ. Already in thediscrete case, the complexity of a problem usually depends heavily on the chosenencoding; e.g., numbers in unary versus binary. This dependence becomes evenmore important when dealing with objects from a continuum like the set of realsand their computability. While Recursive Analysis usually considers one particu-lar encoding for R, the Type-2 Theory of Effectivity (TTE) due to Weihrauchprovides (a convenient formal framework for studying and comparing) a varietyof encodings for different universes. Formally speaking, a representation α for Ris a partial† surjective mapping α :⊆ Σω → R; and an infinite string σ ∈ dom(α)is regarded as an α–name for the real number x = α(σ).

In this way, (α → β)–computing‡ a real function f : R → R means tocompute a transformation on infinite strings F :⊆ Σω → Σω such that any α–name σ for x = α(σ) gets transformed to a β–name τ = F (σ) for f(x) = y, thatis, satisfying β(τ ) = y; cf. [Wei01, Section 3]. Converting α–names to β–namesthus amounts to (α → β)–computability of id : R → R, x 7→ x, and is calledreducibility “α � β” [Wei01, Definition 2.3.2]. Computational equivalence, that

† indicated by the symbol “⊆”, whose absence here generally refers to total functions‡ We use this notation instead of [Wei01]’s (α, β)–computability to stress its connection(but not to be confused) with [α→β]–computability appearing in Section 4.2.

Real Hypercomputation and Continuity 5

is mutual reducibility α � β and β � α, is denoted by “α ≡ β” whereas “α ¬ β”means α � β but β 6� α.

We borrow from TTE also two ways of constructing new representationsfrom giving ones: The conjunction α ∧ α of α and α is the least upper boundwith respect to “�” [Wei01, Lemma 3.3.8]; and for (finitely or countably many)representations αi :⊆ Σω → Ai, their product

∏

i αi denotes a natural represen-tation for the set

∏

iAi [Wei01, Definition 3.3.3.2]. In particular, in order toencode x ∈ R as a rational sequence (qn) ∈ Qω, we (often implicitly) refer tothe representation [νQ]

ω :⊆ Σω → Qω due to [Wei01, Definition 3.1.2.4 andLemma 3.3.16].

2.2 Arithmetic Hierarchy of Real Representations

Observe that (the characterizations due to Fact 3 of) each level of the RealArithmetic Hierarchy gives rise not only to a notion of computability for realnumbers but also canonically to a representation for R; for instance let

ρρρ : encode (arbitrary!) x ∈ R as a fast convergent rational sequence (qn);

ρ<ρ<ρ< : encode x ∈ R as a rational sequence (qn) with supremum x = supn qn;

ρ′ρ′ρ′ : encode x ∈ R as a rational sequence (qn) with limit x = limn qn;

ρ′<ρ′<ρ′< : encode x ∈ R as (qn) ⊆ Q with x = supi infj q〈i,j〉;

ρ′′ρ′′ρ′′ : encode x ∈ R as (qn) ⊆ Q with x = limi limj q〈i,j〉.

As already pointed out, the first three of them are already known and usedin TTE as ρ, ρ<, and ρCn, respectively [Wei01, Section 4.1]. In general oneobtains, similar to Definition 2, a hierarchy of real representations as follows:

Definition 4. Let ρ(0) := ρ, ρ(0)< := ρ<, ρ

(0)> := ρ>. Now fix 1 ≤ d ∈ N:

A ρ(d)–name for x ∈ R is (a [νQ]ω–name for) a rational sequence (qn) such that

x = limn1

limn2

. . . limnd

q〈n1,...,nd〉 .

A ρ(d)< –name for x ∈ R is a (name for a) sequence (qn) ⊆ Q such that

x = supn1

infn2

. . . Θnd+1

q〈n1,...,nd+1〉 .

A ρ(d)> –name for x ∈ R is a sequence (qn) ⊆ Q such that x = inf

n1

supn2

. . .

Regarding Fact 3, one may see ρ′ and ρ′′ as the first and second Jump of ρ,respectively; same for ρ′< and ρ<.

Results from [ZW01] about the Real Arithmetic Hierarchy are easily re-phrased in terms of these representations. Fact 3d) for example translates asfollows:

x is ρ(d)–computable iff it is both ρ(d)< –computable and ρ

(d)> –computable.

Observe that this is a non-uniform claim whereas closer inspection of the proofsin particular of Lemma 3.2 and Lemma 3.3 in [ZW01] reveals them to holdfully uniformly so that we have

6 Martin Ziegler

Lemma 5. ρ ≡ ρ<∧ ρ> ¬ ρ< ¬ ρ′ ≡ ρ′<∧ ρ′> ¬ ρ′< ¬ ρ′′ ≡ . . . .

Moreover, the uniformity of [ZW01, Lemma3.2] yields the following interesting

Scholium§ 6 Let ρ′< denote the representation encoding x ∈ R as (qn) ⊆ Q withx = lim infn qn; and ρ′< similarly with the additional requirement that qn < x forinfinitely many n.Then it holds ρ′< ≡ ρ′< ≡ ρ′< (ρ′< � ρ′< � ρ′< being the trivial direction).

3 Computability and Continuity

Recursive Analysis has established as folklore that any computable real functionis continuous. More precisely, computability of a partial function from/to infinitestrings f :⊆ Σω → Σω requires continuity with respect to the Cantor TopologyτC [Wei01, Theorem 2.2.3]; and this requirement carries over to functions f :⊆A → B on other topological spaces (A, τA) and (B, τB) where input a ∈ A andoutput b = f(a) are encoded by respective admissible representations α and β.Roughly speaking, this property expresses that the mappings α :⊆ Σω → Aand β :⊆ Σω → B satisfy a certain compatibility condition with respect to thetopologies τA/τB and τC involved. For A = B = R, the (standard) representationρ for example is admissible [Wei01, Lemma 4.1.4.1], thus recovering the folkloreclaim.

Now in order to treat and non-trivially investigate computability also of dis-continuous real functions f : R → R, there are basically two ways out: Eitherenhance the underlying Type-2 Machine model or resort to non-admissible rep-resentations. It turns out that for either choice, at least the straight-forwardapproaches fail:• extending Turing Machines with oracles as well as• considering weakened representations for R.

3.1 Type-2 Oracle Computation

Specifically concerning the first approach, most results in Computable Analysisrelativize. Specifically we make

Observation 7. Let O ⊆ Σ∗ be arbitrary. Replace in [Wei01,Definition 2.1.1]the Turing Machine M by MO, that is, one with oracle access to O. This Type-2Computability in O still satisfies

a) closure under composition [Wei01, Theorem 2.1.12];b) computability of string functions requires continuity [Wei01,Theorem 2.2.3];c) same for computable functions on represented spaces with respect to admis-

sible representations [Wei01, Corollary 3.2.12].

§ A scholium is “a note amplifying a proof or course of reasoning, as in mathematics”[Mor69]

Real Hypercomputation and Continuity 7

In particular, the Main Theorem of Recursive Analysis [Wei01, Theorem 4.3.1]relativizes.

A strengthening and counterpart to Observation 7b), we have

Lemma 8. For a partial function on infinite strings f :⊆ Σω → Σω, thefollowing are equivalent:

• There exists an oracle O such that f is computable relative to O;• f is Cantor-continuous and dom(f) is a Gδ–set.

Compare this with Type-1 Theory (that is, computability on finite strings) whereevery function f :⊆ Σ∗ → Σ∗ is recursive in some appropriate O ⊆ Σ∗.

Proof (Lemma 8). If f is recursive in O, then it is also continuous by Observa-tion 7b), that is, the relativized version of [Wei01, Theorem 2.2.3]. Furthermorethe relativization of [Wei01, Theorem 2.2.4] reveals dom(f) to be a Gδ–set.

Conversely suppose that continuous f has Gδ domain. Then f = hω for somemonotone total function h : Σ∗ → Σ∗ according to [Wei01, Theorem 2.3.7.2]where, by [Wei01, Definition 2.1.10.2], hω :⊆ Σω ∋ σ 7→ supn h(σ1 . . . σn)denotes the (existing and unique) extension of h from Σ∗ to ⊆ Σω. A classicalType-1 function on finite strings, this h is recursive in a certain oracle O ⊆ Σ∗.The relativization of [Wei01, Lemma 2.1.11.2] then asserts also hω = f to becomputable in O. ⊓⊔The conclusion of this subsection is that oracles do not increase the computa-tional power of a Type-2 Machine sufficiently in order to handle also discontin-uous functions. So let us proceed to the second approach to real hypercomputa-tion:

3.2 Weaker Representations for Reals

In the present section we are interested in relaxations of the standard repre-sentation ρ for single reals and their effect on the computability of functionevaluation x 7→ f(x). Since, with exception of ρ, none of the ones introduced inDefinition 4 is admissible with respect to the usual Euclidean¶ topology on R[Wei01, Lemma 4.1.4, Example 4.1.14.1], the relativized Main Theorem (Ob-servation 7c) is not applicable. Hence, chances are good for evaluation x 7→ f(x)to become computable even for discontinuous f : R → R; and indeed we havethe following

Example 9. Heaviside’s function

h : R → R, x 7→ 0 for x ≤ 0, x 7→ 1 for x > 0

is both (ρ< → ρ<)–computable and (ρ′< → ρ′<)–computable.

Proof. Given (qn) ⊆ Q with x = supn qn, exploit (νQ → νQ)–computability ofthe restriction h|Q : Q → {0, 1} to obtain pn := h(qn). Then indeed, (pn) ⊆ Q¶ it might be admissible w.r.t. some other, typically artificial topology, though

8 Martin Ziegler

has supn pn = h(x): In case x ≤ 0, qn ≤ 0 and hence pn = 0 for all n; whereasin case x > 0, qn > 0 and hence pn = 1 for some n.

Let x ∈ R be given by a rational double sequence (qi,j) with x = supi infj qi,j .Proceeding from qi,j to qi,j := max{q0,j, . . . , qi,j}, we assert infj qi+1,j ≥ infj qi,j .Now compute pi,j := h(qi,j − 2−i). Then in case x ≤ 0, it holds ∀i∃j : qi,j ≤ 2−i,i.e., pi,j = 0 and thus supi infj pi,j = 0 = h(x). Similarly in case x > 0, there issome i0 such that infj qi0,j > x/2 and thus infj qi,j > x/2 for all i ≥ i0. For i ≥ i0with 2−i ≤ x/2, it follows pi,j = 1 ∀j and therefore supi infj pi,j = 1 = h(x). ⊓⊔So real function hypercomputation based on weaker representations indeed doesallow for effective evaluation of some discontinuous functions. On the other hand,they still impose well-known topological restrictions:

Fact 10. Consider f : R → R.

a) If f is (ρ → ρ)–computable, then it is continuous.b) If f is (ρ → ρ<)–computable, then it is lower semi-continuous.c) If f is (ρ< → ρ<)–computable, then it is monotonically increasing.d) If f is (ρ′ → ρ′)–computable, then it is continuous.

The claims remain valid under oracle-supported computation.

Claim a) is the Main Theorem. For b) see [WZ00] and recall, e.g. from [Ran68,Chapter 6.7], that f : R → R is lower semi-continuous iff f(limn xn) ≤lim infn f(xn) for all convergent sequences (xn); equivalently: f−1

[

(y,∞)]

isopen for any y ∈ R. The establishing of d) in [BH02, Section 6] caused somesurprise. We briefly sketch the according proofs as a preparation for those ofTheorem 11 below.

Proof. a) Suppose for a start that Heaviside’s function, in spite of its disconti-nuity at x = 0, be (ρ → ρ)–computable by some Type-2 Machine M. Feedthe rational sequence qn := 2−n, a valid ρ–name for x, to this M. By pre-sumption it will then spit out a sequence (pm)

m⊆ Q with |pm − y| ≤ 2−m

for y = h(x) = 0; in particular, |p2 − y| > 2−2 for y := 1. Up to output ofp2, M has executed a finite number N ∈ N of operations and in particularread at most the initial part p0, p1, . . . , pN of the input.Now re-use M in order to evaluate h at x := pN > 0 ρ–encoded as therational sequence (qn) := (q0, q1, . . . , qN , qN , . . .) coinciding with (pn) forn ≤ N . Being a deterministic machine, M will then proceed exactly asbefore for its first N steps; in particular the output (pm) agrees with (pm)up to m = 2. Hence |p2 − y| > 2−2 contradicting that M is supposed tooutput a ρ–name for y = h(x).

For the case of a general function f : R → R with discontinuity at somex ∈ R, let y = f(x) 6= limk f(xk) = y with a real sequence xk convergingto x. There exists M ∈ N with |y − y| > 2−M+2; by possibly proceeding toan appropriate subsequence of (xk), we may suppose w.l.o.g. that |x−xk| ≤2−k−2 and |f(xk)−y| ≤ 2−M . Then there is a rational double sequence (qk,n)such that |xk − qk,n| ≤ 2−n−1; thus |x− qn,n| ≤ 2−n. We may therefore feed

Real Hypercomputation and Continuity 9

(qn,n) as a ρ–name in order to evaluate f at x and obtain in turn a ρ–name(pm) ⊆ Q for y. As before, pM is output after having only read some finiteinitial part (qn,n)n≤N

of the input. Then

|qn,n−xN | ≤ |qn,n−xn|+|xn−x|+|x−xN | ≤ 2−n−1+2−n−2+2−N−2 ≤ 2−n

for n ≤ N reveals this very initial part to also be the start of a valid ρ–namefor x := xN whereas

2−M+2 < |y−y| ≤ |y−pM |+|pM−f(x)|+|f(x)−y| ≤ 2−M+|pM−f(x)|+2−M

shows that (pm)m≤M is not a valid initial part of a ρ–name for f(x): con-tradiction.

b) We prove (ρ → ρ<)–uncomputability of the flipped Heaviside Function

h : 0 ≥ x 7→ 1, 0 < x 7→ 0

as a prototype lacking lower semi-continuity.Consider again the ρ–name qn := 2−n for x = 0 which the hypotheticalType-2 Machine transforms into a ρ<–name for y = h(x) = 1, that is, asequence (pm) ⊆ Q with supm pm = y; In particular pM ≥ 2

3 for someM ∈ N gets output having read only (qn)n≤N

for some N ∈ N. The latterfinite segment is also the initial part of a valid ρ–name for x = qN > 0whereas (pm)

m≤Mhas sup ≥ 2

3 and thus is not the initial part of a valid

ρ<–name for y = h(x) = 0: contradiction.This proof for the case h carries over to an arbitrary f : R → R just likein a), that is, by replacing qn = 2−n with rational approximations to ageneral sequence xn ∈ R witnessing violated lower semi-continuity of f inthat f(limn xn) > lim infn f(xn).

c) As in a) and b), we treat for notational simplicity the case of f : R → Rviolating monotonicity in that f(0) = 1 and f(1) = 0; the general case canagain be handled similarly. Feed the ρ<–name (qn) = (0, 0, . . .) for x = 0into a machine which be presumption produces a sequence (pm) ⊆ Q withsup pm = 1 and in particular pM ≥ 2

3 for some M ∈ N. Up to output ofpM , only (qn)n≤N

has been read for some N ∈ N. Now consider the rationalsequence (qn) consisting of N zeros followed by an infinity of 1s, that is, avalid ρ<–name for x = 1. This new input will cause the machine to output asequence (pm) ⊆ Q coinciding with (pm) for m ≤ M ; in particular pM ≥ 2

3contradicting that (pm) is supposed to satisfy supm pm = f(x) = 0.

d) Suppose that, in spite of its discontinuity at x = 0, h be (ρ′ → ρ′)–computable by some Type-2 Machine M.

Consider the sequence q(1) :=(

q(1)n

)

⊆ Q, q(1)n :≡ 1, which is by definition a

valid ρ′–name for 1 =: x(1) = limn q(1)n . So upon input of q(1), M will gener-

ate a corresponding sequence p(1) ⊆ Q as a ρ′–name for y(1) = h(

x(1))

= 0,

that is, satisfying limm p(1)m = 0; in particular, p

(1)m1

≤ 13 for some m1 ∈ N. Up

to this output, M has read only a finite initial part of the input q(1), say,up to n ≤ n1.

10 Martin Ziegler

Next consider the sequence q(2) ⊆ Q defined by q(2)n := 1 for n ≤ n1 and

q(2)n := 1

2 for n > n1: a valid ρ′–name for x(2) = 12 which M by presumption

transforms into a sequence p(2) ⊆ Q with limm p(2)m = y(2) = h

(

x(2))

= 0; in

particular, q(2)m2

≤ 13 for some m2 > m1. However, due to M’s deterministic

behavior and since q(1) and q(2) initially coincide, it still holds p(2)m1 ≤ 1

3 .Now by repeating the above argument we obtain a sequence of sequencesq(k) ⊆ Q, each constant for n ≥ nk of value (and thus a valid ρ′–namefor) x(k) = 2−k+1 and transformed by M into a sequence p(k) ⊆ Q satisfying

p(k)mi ≤ 1

3 for i = 1, . . . k with strictly increasing (nk), (mk) ⊆ N. The ultimate

sequence q(ω) ⊆ Q, well-defined by q(ω)n := q

(k)n for n ≤ nk (and in fact the

limit of the sequence of sequences(

q(k))

kwith respect to Baire’s Topology),

therefore converges to (and is hence a valid ρ′–name for) x(ω) = 0; and it

gets mapped by M to a sequence q(ω) ⊆ Q satisfying q(ω)m ≤ 1

3 for infinitely

many m contradicting that a valid ρ′–name for y(ω) = h(

x(ω))

= 1 shouldhave limm = 1.

Being only information-theoretic, the above arguments obviously relativize. ⊓⊔The main result of the present section is an extension of Fact 10 to one level upon the hierarchy of real representations from Definition 4. This suggests similarclaims to hold for the entire hierarchy and might not be as surprising any moreas Fact 10d) in [BH02]; nevertheless, already this additional step makes proofssignificantly more involved.

Theorem 11 (First Main Theorem of Real Hypercomputation).Consider f : R → R.

a) If f is (ρ′ → ρ′<)–computable, then it is lower semi-continuous.b) If f is (ρ′< → ρ′<)–computable, then it is monotonically increasing.c) If f is (ρ′′ → ρ′′)–computable, then it is continuous.

The claims remain valid under oracle-supported computation.

We point out that the proofs of Fact 10 proceed by constructing an input forwhich a presumed machine M fails to produce the correct output. They differhowever in the ‘length’ of these constructions: for Claims a) to c), the counter-example inputs are obtained by running M for a finite number of steps on asingle, fixed argument; whereas in the proof of Claim d), M is repeatedly startedon an adaptively extended sequence of arguments. The latter argument may thusbe considered as of length ω, the first infinite ordinal. Our proof of Theorem 11c)will be even longer and is therefore put into the following subsection.

3.3 Proof of Theorem 11

As in the proof of Fact 10, we treat the special case of the flipped HeavisideFunction h for reasons of notational convenience and clarity of presentation; theaccording arguments can be immediately extended to the general case.

Real Hypercomputation and Continuity 11

Claim 12. h : R → R is not (ρ′ → ρ′<)–computable.

Proof. Suppose a Type-2 Machine M (ρ′ → ρ′<)–computes h. In particular,

upon input of x(1) = 1 in form of the sequence q(1) = (q(1)n ) with q

(1)n :≡ 1, M

will output a rational double sequence p(1) = (p(1)k,ℓ) with 0 = y(1) := h(x(1)) =

supk

infℓp(1)k,ℓ. Observe that p

(1)1,ℓ1

≤ 13 for some ℓ1. When writing p

(1)1,ℓ1

, M has only

read a finite part of (q(1)n ), say, up to n1.

Now consider x(2) := 12 , given by way of the sequence q(2) with q

(2)n := 1

for n < n1 and q(2)n := 1

2 for n ≥ n1. Then, too, M will output a double

sequence p(2) with 0 = y(2) = supk

infℓp(2)k,ℓ. Observe that, similarly, some p

(2)2,ℓ2

≤ 13

is output having read only a finite part of (q(2)n ), say, up to n2. Moreover, as q(1)

and q(2) coincide up to n1 and sinceM operates deterministically, p(2)1,ℓ1

= p(1)1,ℓ1

≤13 .

Fig. 1. Illustration to the iterative construction employed in the proof ofClaim 12

Continuing this process with x(k) := 2−k+1 for k = 3, 4, . . . as indicated in

Figure 1 eventually yields a rational sequence q(ω) with limn q(ω)n =: x(ω) = 0,

upon input of which M outputs a double sequence p(ω) such that p(ω)k,ℓk

≤ 13 for

all k = 1, 2, . . .. In particular, y(ω) := supk

infℓp(ω)k,ℓ ≤ 1

3 whereas h(x(ω)) = 1 :

contradiction. ⊓⊔

Notice that the above proof involves one-dimensionally indexed sequences (qn)for input and two-dimensionally indexed ones (pk,ℓ) for output. We now proceeda step further in proof difficulty, namely involving two-dimensional indices forboth input and output in order to establish Item b).

Claim 13. Let f : R → R violate monotonicity in that f(0) = 1 and f(1) = 0.Then, f is not (ρ′< → ρ′<)–computable.

Proof. We construct a ρ′<–name for x = 0 from an iteratively defined sequenceof initial segments of ρ′<–names for x = 1:

12 Martin Ziegler

Start with q(1)i,j := 1 for all i, j. Then, q(1) = (q

(1)i,j ) is obviously a ρ′<–name

for x = 1 and thus yields by presumption, upon input to M, a ρ′<–name p(1)k,ℓ for

f(1) = 0, that is, with 0 = supk

infℓp(1)k,ℓ. In particular, p

(1)1,ℓ1

≤ 13 for some ℓ1.

Fig. 2. Illustration to the iterative construction employed in the proof ofClaim 13

Until output of p(1)1,ℓ1

, M has read only finitely many entries of q(1); say, up toi1 and j1, that is, covered in Figure 2 by the light gray rectangle. Now consider

q(2) defined as in this figure: Since infj q(2)i,j = 0 for i ≤ i1 and infj q

(2)i,j = 1

for i > i1, supi

infjq(2)i,j = 1, that is, this is again valid ρ′<–name for x = 1; and

again, M will by presumption convert q(2) into a ρ′<–name p(2) for f(1) = 0.

In particular, p(2)2,ℓ2

≤ 13 for some ℓ2; and, being a deterministic machine, M’s

operation on the initial part (dark gray) on which input q(2) coincides with input

q(1) will first have generated the same initial output, namely p(2)1,ℓ1

= p(1)1,ℓ1

≤ 12 .

Again, until output of p(2)2,ℓ2

, M has read only a finite part of q(2) of, say, up to

i2 > i1 (light gray). By now considering input q(3) with infj q(3)i,j = 0 for i ≤ i2

as in Figure 2, we arrive at p(3) and ℓ3 with p(3)1,ℓ1

, p(3)2,ℓ2

, p(3)3,ℓ3

≤ 13 ; and so on with

i3, q(4), p(4), ℓ4, i4, . . .

Finally observe that continuing these arguments eventually leads to a rational

double sequence q(ω) = (q(ω)i,j ) which has infj q

(∞)i,j = 0 for i ≤ i∞ = ∞—and

is therefore a valid ρ′<–name for x = 0 (rather than x = 1)—but gets mapped

by M to p(ω) = (p(ω)k,ℓ ) with infℓ p

(∞)k,ℓ ≤ p

(∞)k,ℓk

≤ 13 for all k. Since f(0) = 1,

this contradicts our presumption that M maps ρ′<–names for x to ρ′<–names forf(x). ⊓⊔

The above proofs involving ρ′ and ρ′< proceeded by constructing an infinitesequence of inputs q(1), q(2), . . . , q(ω) (each possibly a multi-indexed sequence of

Real Hypercomputation and Continuity 13

its own). For finally asserting Claim c) involving ρ′′, we will extend this methodfrom length ω, the first infinite ordinal, to an even longer one.

Claim 14. h : R → R is not (ρ′′ → ρ′′)–computable.

Proof. Outwit a Type-2 Machine M, presumed to realize this computation, asfollows:

i) Take q(1) to be the constant double sequence 1, i.e., q(1)i,j := 1 for all i, j.

Being a ρ′′–name for 1, it is by presumption mapped to a ρ′′–name p(1) for

h(1) = 0, that is, satisfying limk limℓ p(1)k,ℓ = 0. In particular, almost every

column #k contains an entry #ℓ with p(1)k,ℓ ≤ 1

3 . Until output of the first

such p(1)k1,ℓ1

, M has read only a finite part of q(1)—say, up to i1, j1.

Fig. 3. The first infinitely long iterative construction employed in the proof ofClaim 14

ii) Observe that this Argument i) equally applies to the scaled input sequence

2−m · q(1) for any m. So define q(2)i,j := q

(1)i,j for j ≤ j1 (i.e., inherit the

initial part of q(1)) and q(2)i,j :≡ 1

2 for j > j1. Now upon input of this q(2),

M will output p(2) with, again, infinitely many p(2)k,ℓ ≤ 1

3 , the first one—

(k2, ℓ2), say—after having read q(2) only up to some (i2, j2). Furthermore

M’s determinism implies p(2)k1,ℓ1

= p(1)k1,ℓ1

≤ 13 .

By repeating for m = 2, 3, . . ., we eventually obtain—similarly to the proof

of Claim 13—an input sequence q(ω) with q(ω)i,j with limi limj q

(ω)i,j = 0, that

is, a valid ρ′′–name for x = 0 (rather than 1). This is mapped by M to p(ω)

with p(ω)km,ℓm

≤ 13 for all m. On the other hand, p(ω) is by presumption a

ρ′′–name for h(0) = 1. Therefore, there are infinitely many m with p(ω)m,ℓ ≥ 2

3

for some ℓ > ℓm and p(ω)m,ℓm

≤ 13 ; see the grey columns in the right part of

Figure 3.

14 Martin Ziegler

iii) Since this gives no contradiction yet, we proceed by considering the firstsuch column m containing an entry ≤ 1

3 as well as an entry ≥ 23 . Take the

initial part of the input q(ω) — up to (iω, jω), say, depicted in grey in the leftpart of Figure 4 — that M has read until output of both of them; extendit with 1

2 s in top direction and with 1s to the right. Feed this ρ′′–name forx = 1 into M until output of an entry pk,ℓ ≤ 1

3 in some column k beyondm. Then repeat extending to the right with 1s replaced by 1

2 s for a secondentry pk,ℓ ≤ 1

3 .

Fig. 4. Second infinitely long iterative construction employed in the proof ofClaim 14

More generally, proceed similarly as in ii) and extend q(ω)iω ,iω

to the right in

such a way with some ρ′′–name q′(ω) for x = 0 as to obtain another columnm′ with both entries ≤ 1

3 and ≥ 23 ; see the middle part of Figure 4. Again,

M outputs the latter two entries having read only a finite part; say, up to(i′ω, j

′ω).

Now extend this part, too, with 12 in top direction and with another q′′(ω)

obtained, again, as in ii) for a third column m′′ with both entries ≤ 13 and

≥ 23 ; and so on.

This eventually leads to an input q(2ω) which, due to the extensions to thetop, represents a ρ′′–name for x = 1

2 and is thus mapped by presumption to

a ρ′′–name p(2ω) for h(12 ) = 0. In particular, almost every column of p(2ω)

has almost every entry ≤ 13 while maintaining infinitely many columns with

preceding entries ≤ 13 and ≥ 2

3 ; see the right part of Figure 4. This asserts

the existence of infinitely many columns in p(2ω) containing ≤ 13 , ≥ 2

3 , and

≤ 13 in order. And again, already a finite initial part of q(2ω) up to some

(i2ω, j2ω) gives rise to the first such triple.iv) Notice that the arguments in iii) similarly yield the existence of an appro-

priate, scaled counter-part 12q

′(2ω) of q(2ω), of some 14q

′′(2ω), and so on, allleading to output containing infinitely many columns with alternating triples

Real Hypercomputation and Continuity 15

as above. We now construct input q(3ω) leading to output p(3ω) containingan infinity of columns, each with four entries ≤ 1

3 , ≥ 23 , ≤ 1

3 , and ≥ 23 .

To this end, take the initial part of q(2ω) leading to output of the first columnwith alternating triple in the above sense; then extend it with the initial partof the scaled version 1

2q′(2ω) leading to another column with such a triple; and

so on. Observing that, due to the scaling, the thus obtained q(3ω) representsa ρ′′–name for x = 0, almost every column of the output p(3ω) representingh(0) = 1 contains entries ≥ 2

3 in addition to the infinitely many columnswith triples as above; see the left part of Figure 5.

Fig. 5. Third, fourth, and fifth infinitely long iterative construction employed inthe proof of Claim 14

v) Our next step is a ρ′′–name q(4ω) for x = 14 giving rise to p(4ω) with in-

finitely many columns containing alternating quintuples. This is obtainedby repeating the arguments in iv) to obtain initial segments of (variants of)q(3ω), stacking them horizontally—in order to obtain an infinity of columnswith alternating quadruples—while extending in top direction with 1

4 ; see

the middle part of Figure 5. This forces M to output a ρ′′–name q(4ω) forh(14 ) = 0 and thus with in almost every column almost every entry being≤ 1

4 , thus extending the alternating quadruples to quintuples.vi) Noticing that the vertical extension in v) was similar to step iii), we now

take a step similar to iv) based on horizontally stacked initial parts of scaledcounterparts of q(4ω) in order to obtain a ρ′′–name q(5ω) for x = 0 which Mmaps to some p(5ω) containing infinitely many alternating six-tuples.Then again construct a ρ′′–name q(6ω) for x = 1

8 by horizontally stacking

initial segments of (variants of) q(5ω) while extending them vertically with18 and so on.

Now for the bottom line: By proceeding the above construction, one eventually

obtains a rational double sequence q(ω2) with limj q

(ω2)i,j = 0 for all i — that

16 Martin Ziegler

is, a ρ′′–name for x = 0 — mapped by M to some p(ω2) containing (infinitely

many) columns #k with infinitely many alternating entries ≤ 13 and ≥ 2

3 —contradicting that, for ρ′′-names p = (pk,ℓ), limℓ pk,ℓ is required to exist forevery k. ⊓⊔

4 Hierarchies of Hypercomputable Real Functions

The present section investigates and compares the first levels of the two hierar-chies of hypercomputable real functions induced by the two approaches to realfunction hypercomputation considered in Section 3: based on oracle support andbased on weakened encodings.

4.1 Weakly Computable Real Functions

For every (α → β)–computable function f : A → B, one may obviously replacerepresentation α for A by a stronger one and β for B by a weaker one whilemaintaining computability of f :

f (α → β)–computable ∧ α,� α ∧ β � β′ ⇒ f (α,→ β′)–computable.

However if both α and β are made, say, weaker then (α′ → β′)–computabilityof f may in general be violated. For α = β = ρ<, though, we have seen inExample 9 that the implication “(ρ< → ρ<) ⇒ (ρ′< → ρ′<)” does hold at leastfor the case of f being Heaviside’s function. By the following result, it holds infact for every f :

Theorem 15 (Second Main Theorem of Real Hypercomputation).Consider f : R → R.

a) If f is (ρ → ρ)–computable, then it is also (ρ′ → ρ′)–computable.b) If f is (ρ → ρ<)–computable, then it is also (ρ′ → ρ′<)–computable.c) If f is (ρ< → ρ<)–computable, then it is also (ρ′< → ρ′<)–computable.d) If f is (ρ′ → ρ′)–computable, then it is also (ρ′′ → ρ′′)–computable.e) If f is (ρ′′ → ρ′′)–computable, then it is also (ρ′′′ → ρ′′′)–computable.

The claims remain valid under oracle-supported computation.

As a consequence, we obtain the following partial strengthening of Lemma 5:

Corollary 16. It holds ρ ≡ ρ<∧ρ> ¬t ρ< ¬t ρ′ ≡ ρ′<∧ρ′> ¬t ρ′< ¬t ρ′′

where “�t” denotes continuous reducibility of representations [Wei01,Def. 2.3.2].

Proof. The positive claims follow from Lemmas 5 and 8. For a negative claimlike “ρ′< �t ρ

′” suppose the contrary. Then by Lemma 8, with the help of someappropriate oracle O, one can convert ρ′<–names to ρ′–names. As Heaviside’sfunction h is (ρ′ → ρ′<)–computable by Example 9 and Theorem 15, compositionwith the presumed conversion implies (ρ′ → ρ′)–computability of h relative toO—contradicting Theorem 11c). ⊓⊔

Real Hypercomputation and Continuity 17

Proof (Theorem 15d). Let f be (ρ′ → ρ′)–computable and x given by a ρ′′–name, that is, a rational sequence q = (qn) with x = limi limj q〈i,j〉. For each i,compute by assumption from the ρ′–name q〈i,·〉 = (q〈i,j〉)j of xi := limj q〈i,j〉 aρ′–name of f(xi), that is, a sequence p = p〈i,·〉 = (p〈i,j〉)j with f(xi) = limj p〈i,j〉.Continuity of f due to Fact 10c) asserts

limilimj

p〈i,j〉 = limif(xi)

!= f

(

limi

xi

)

= f(

limilimj

q〈i,j〉

)

= f(x)

this sequence p to be a ρ′′–name for y = f(x). ⊓⊔Where the last proof exploited Fact 10c), the next one relies on Theorem 11c):

Proof (Theorem 15e). A ρ′′′–name for x ∈ R is a rational sequence a = (qn) withx = limi limj limk q〈i,j,k〉. For each i, exploit (ρ′′ → ρ′′)–computability of f toobtain, from the ρ′′–name q〈i, · , · 〉 of xi := limj limk q〈i,j,k〉 ∈ R, a sequence p〈i, · , · 〉

with limj limk p〈i,j,k〉 as ρ′′–name of f(xi). Similarly to case d), this sequence pconstitutes a ρ′′′–name for y = f(x) by continuity of f due to Theorem 11c). ⊓⊔Proof (Theorem 15a). Let f be (ρ → ρ)–computable. Its (ρ′ → ρ′)–computabilityis established as follows: Given (qn) ⊆ Q with x = limn qn, apply the assump-tion to evaluate f(qn) for each n up to error 2−n; that is, obtain pn ∈ Qwith |pn − f(qn)| ≤ 2−n. Since f is continuous by Fact 10a), it follows f(x) =limn f(qn) = limn pn so that (pn) is a ρ′–name for y = f(x). ⊓⊔It is interesting that the latter proof works in fact uniformly in f , i.e., we have

Scholium 17. The apply operator C(R)×R ∋ (f, x) 7→ f(x) is ([ρ→ρ]×ρ′ →ρ′)–computable.

Similarly, Theorem 15b) follows from Lemma 18 below together with the ob-servation that every (ρ → ρ<)–computable f has a computable [ρ→ ρ<]–name[WZ00, Corollary 5.1(2) and Theorem 3.7]; here, [ρ→ ρ<] denotes a nat-ural representation for the space LSC(R) of lower semi-continuous functionsf : R → R considered in [WZ00]. Specifically, a [ρ→ρ<]–name for such a f is anenumeration of all rational triples (a, b, c) such that c < min f

[

[a, b]]

—the lattermaking sense as a lower semi-continuous function attains its minimum (thoughnot necessarily its maximum) on any compact set. [ρ→ρ<] indeed is a represen-tation for LSC(R) because different lower semi-continuous functions give rise todifferent such collections {(a, b, c) ∈ Q3 : . . .}; cf. [WZ00, Lemma 3.3].

Lemma 18. LSC(R)×R ∋ (f, x) 7→ f(x) is ([ρ→ρ<]×ρ′ → ρ′<)–computable.

Proof. Let (ak, bk, ck)k denote the given [ρ→ρ<]–name of f ∈ LSC(R) and (qn)nthe given ρ′–name for x ∈ R. Our goal is to ρ′<–compute y := f(x). Define thesequence p = (pm)

m⊆ Q ∪ {+∞} by

p〈k,ℓ,n〉

:=

max{

cm : m ≤ k ∧ [am, bm] ⊇ [ak, bk]}

if qn ∈ (ak, bk) ∧ |bk − ak| = 2−ℓ

+∞ otherwise

(1)

From the given information, one can obviously compute p. Moreover this se-quence satisfies

18 Martin Ziegler

• lim inf p ≥ y:Let ǫ > 0 be arbitrary. Since f is lower semi-continuous, its preimagef−1

[

(y − ǫ,∞)]

∋ x is an open set and therefore contains an entire ballaround x. In fact, the center of this ball may be chosen as rational and itsdiameter of the form 2−L for some L ∈ N; formally (see Figure 6):

∃K,L,K ′ ∈ N : x ∈ (aK′ , bK′) ⊆ [aK , bK ] ⊆ f−1[

(y − ǫ,∞)]

∧|bK − aK | = 2−L ∧ aK′ = aK + 3

2 · 2−L−2 ∧ bK′ = bK − 32 · 2−L−2 (2)

where we have exploited that every rational pair (a, b) occurs in the listrepresenting the [ρ→ρ<]–name. Moreover, as it consists of all rational triples(a, b, c) with c < min f

[

[a, b]]

,

∃M ≥ K : [aK , bK ] = [aM , bM ] ∧ cM ≥ min f[

[aM , bM ]]

− ǫ(*)

≥ y−2ǫ (3)

with (*) a consequence of [aK , bK ] ⊆ f−1[

(y−ǫ,∞)]

in Equation (2). Finally,

limn

qn = x ∈ (aK′ , bK′) ⇒ ∃N : ∀n ≥ N : qn ∈ (aK′ , bK′) . (4)

So putting things together, for each n ≥ N , ℓ ≥ L′, and k ≥ M , we eitherhave p

〈k,ℓ,n〉= +∞ ≥ y− 2ǫ; or we are in the first case of Equation (1), thus

– qn ∈ (ak, bk) with |bk − ak| ≤ 2−ℓ

– qn ∈ (aK′ , bK′) by Equation (4)– hence [ak, bk] ⊆ [aK , bK ] by Equation (2) due to ℓ ≥ L′; cf. Figure 6.– So [ak, bk] ⊆ [aM , bM ] by Equation (3)– implying p

〈k,ℓ,n〉≥ cM ≥ y − 2ǫ by Equations (1) and (3) since k ≥ M .

Summarizing, it holds p〈k,ℓ,n〉

≥ y − 2ǫ for all (k, ℓ, n) ∈ N3 not belongingto the finite set {0, 1, . . . , N − 1} × {0, 1, . . . , L′ − 1} × {0, 1, . . . ,M − 1} ofexceptions. Consequently lim inf p ≥ y− 2ǫ; even lim inf p ≥ y because ǫ > 0was arbitrary.

Fig. 6. Nesting of some rational intervals of dyadic length contained in f−1[

(y−ǫ,∞)

]

.The parameters are chosen in such a way that, whenever (aK′ , bK′) meets someother (ak, bk) of length |bk−ak| = 2−ℓ for ℓ ≥ L′ := L+2, then [ak, bk] is entirelycontained within the larger [aK , bK ].

Real Hypercomputation and Continuity 19

• lim inf p ≤ yIndeed: Since the [ρ → ρ<]–name contains in particular all rational pairs(ak, bk) and these intervals are dense in R, there exists to every ℓ ∈ Nsome k such that |bk − ak| = 2−ℓ and x ∈ (ak, bk). Furthermore it holdsqn ∈ (ak, bk) for some sufficiently large n because limn qn = x. We have thusinfinitely many triples (k, ℓ, n) for which p

〈k,ℓ,n〉is defined by the first case in

Equation (1) and thus agrees with some cm < min f[

[am, bm]]

≤ f(x) = yas x ∈ (ak, bk) ⊆ [am, bm].

Concluding, we have lim infm pm = y. Although p may attain the value +∞,this can easily be overcome by proceeding to pm := pm for pm 6= ∞ andpm := max{0, p0, . . . , pm−1} for pm = ∞ because this transformation p 7→ pon sequences obviously does not affect their lim inf < ∞. This yields a ρ′<–namefor y which can finally be converted to the desired ρ′<–name due to the easy partof Scholium 6. ⊓⊔

In order to obtain a similar uniform claim yielding Theorem 15c), recallthat every (ρ< → ρ<)–computable function f : R → R is necessarily bothmonotonically increasing and lower semi-continuous (Fact 10b+c). This suggests

Definition 19. Let MLSC(R) denote the class of all monotonically increas-ing, lower semi-continuous functions f : R → R. A [ρ< → ρ<]–name for f ∈MLSC(R) is an enumeration of the set {(a, c) ∈ Q2 : c < f(a)}.

Lemma 20. a) Distinct f, g ∈ MLSC(R) have different sets {(a, c) : . . .} ac-cording to Definition 19; that is, [ρ< → ρ<] constitutes a well-defined repre-sentation.

b) A function f ∈ MLSC(R) is (ρ< → ρ<)–computable iff it has a computable[ρ<→ρ<]–name.

c) Let f ∈ MLSC(R), (ak, ck)k with {(a, c) ∈ Q2 : c < f(a)} = {(ak, ck) :k ∈ N}, x ∈ R, and q = (qn) ⊆ Q with x = lim infn qn. Then, the rationalsequence p defined by

p〈k,n,ℓ〉

:=

{

max{

cm : m ≤ k ∧ am ≥ ak}

if ak < qn < ak + 2−ℓ

+∞ otherwise

satisfies lim inf p = f(x) =: y.d) Therefore, the apply operator MLSC(R) × R ∋ (f, x) 7→ f(x) is

([ρ<→ρ<]× ρ′< → ρ′<)–computable.

Proof. a) Let f, g ∈ MLSC(R) with f 6= g, that is, w.l.o.g. f(x0) < g(x0)for some x0 ∈ R. There exists some c0 ∈ Q with f(x0) < c0 < g(x0).Being monotonically increasing and lower semi-continuous, their pre-imagesf−1

[

(c0,∞)]

6∋ x0 and g−1[

(c0,∞)]

∋ x0 on open half-interval (c0,∞) areagain open half-intervals (xf ,∞) and (xg,∞), respectively. As x0 belongsto the second but not to the first, we have xg < x0 < xf and thereforexg < a0 < xf for some a0 ∈ Q. Then a0 ∈ (xg ,∞) = g−1

[

(c0,∞)]

yields

c0 < g(a0) whereas a0 6∈ (xf ,∞) = f−1[

(c0,∞)]

asserts c0 6< f(a0).

20 Martin Ziegler

b) Let M denote a Type-2 Machine (ρ< → ρ<)–computing f ∈ MLSC(R).Evaluating f at a ∈ Q by simulating M on the ρ<–name (a, a, a, . . .) for athus yields a ρ<–name for f(a) which is (equivalent to) a list of all c ∈ Qwith c < f(a) [Wei01, Lemma 4.1.8]. So dove-tailing this simulation for alla ∈ Q yields the desired [ρ<→ρ<]–name for f .Conversely, knowing a [ρ<→ρ<]–name (ak, ck)k for f ∈ MLSC(R) and givenan increasing sequence (qn) ⊆ Q with x = supn qn, let

pn := cn if an ≤ qn, pn := −∞ otherwise .

Then, in the first case, pn = cn < f(an) ≤ f(qn) ≤ f(x) =: y by monotonic-ity, and pn = −∞ ≤ y in the second; hence supn pn ≤ y. To see supn pn ≥ y,fix arbitrary ǫ > 0 and consider the open half-interval f−1

[

(y − ǫ,∞)]

=(xǫ,∞) containing x and thus also some rational a = aK ∈ (xǫ, x), K ∈ N.Furthermore qn ր x yields some N ∈ N such that qn ∈ (aK , x) for alln ≥ N . And finally there exists M ≥ N with aM = aK and cM ≥ f(aM )− ǫ.Together this asserts qM > aK = aM because M ≥ N and thus pM = cM ≥f(aK)− ǫ > y − 2ǫ due to ak ∈ f−1

[

(y − ǫ,∞)]

.c) Take arbitrary ǫ > 0. As f is increasing and lower semi-continuous, the pre-

image f−1[

(y−ǫ,∞)]

is an open half-interval (xǫ,∞) containing x. Thereforethere exist K,L ∈ N such that xǫ < aK and aK +2−L < x; furthermore, thesequence (ak, ck)k containing all rational pairs (a, c) with c < f(a), thereis M ≥ K such that aM = aK and cM ≥ f(aM ) − ǫ; and finally, sincelim infn qn = x > aM + 2−L, it holds qn > aM + 2−L for all n ≥ N with anappropriate N ∈ N. Observe that q > aM +2−L and a < q < a+2−L impliesa ≥ aM ; so together we have for all n ≥ N , ℓ ≥ L, and k ≥ M that p

〈k,n,ℓ〉is

either +∞ or ≥ cM ≥ f(aK) − ǫ ≥ f(xǫ) − ǫ ≥ y − 2ǫ due to monotonicityof f and by definition of xǫ < aK . This proves lim inf p ≥ y because ǫ wasarbitrary.To see the reverse inequality “lim inf p ≤ y”, take arbitrary ℓ ∈ N. Thereexists k ∈ N with ak < x < ak + 2−ℓ and, because of lim infn qn = x, alson ∈ N with ak < qn < ak + 2−ℓ. We therefore have infinitely many triples(n, k, ℓ) for which p

〈n,k,ℓ〉agrees with a certain cm < f(am) ≤ f(ak) ≤ f(x) =

y.d) Given a ρ′<–name for x, one can obtain a sequence (qn) ⊆ Q with x =

lim infn qn by virtue of Scholium 6. From this, the sequence p ⊆ Q withlim inf p = f(x) according to c) is obviously computable and yields, againby Scholium 6, a ρ′<–name for y = f(x). ⊓⊔

Concluding this subsection, the classes of (ρ(d) → ρ(d))–computable real func-tions f : R → R form, for d = 0, 1, . . . respectively, a hierarchy. By Fact 3, thishierarchy is strict as can be seen from the constant functions f(x) ≡ c withc ∈ ∆d+1R.

4.2 Arithmetic Weierstraß Hierarchy

Section 4.1 established the sequence ρ, ρ′, ρ′′, . . . of increasingly weaker repre-sentations for R to yield the strict hierarchy of (ρ → ρ)–computable, (ρ′ → ρ′)–

Real Hypercomputation and Continuity 21

computable, and (ρ′′ → ρ′′)–computable functions f : [0, 1] → R. We now com-pare these classes with those induced by the other kind of real hypercomputationsuggested in Section 3: relative to the Halting Problem H = ∅′ and its iteratedjumps ∅′′, . . .

Such a comparison makes sense because both weakly and oracle-computablereal functions are necessarily continuous according to Fact 10d)/Theorem 11c)and Lemma 8, respectively.

The classical Weierstraß Approximation Theorem establishes any contin-uous real function f : [0, 1] → R to be the uniform limit f = ulimn Pn of asequence of rational polynomials (Pn) ⊆ Q[X ]. Here, ‘ulim’ suggestively denotesuniform convergence of continuous functions on [0, 1], that is the requirement

sup0≤x≤1

|f(x)− Pn(x)| =: ‖f − Pn‖ → 0 as n → ∞ .

The famous Effective Weierstraß Theorem due to Pour-El, Caldwell, andHauck relates effectively evaluable to effectively approximable real functions:

Fact 21. A function f : [0, 1] → R is (ρ → ρ)–computable if and only if it holds

[ρ→ρ][ρ→ρ][ρ→ρ]: There exists a computable sequence of (degrees and coefficients of)

rational polynomials (Pn) ⊆ Q[X ] such that ‖f − Pn‖ ≤ 2−n (5)

Proof. See [PER89, Section 0.7], [PEC75], or [Hau76].The notion “[ρ→ ρ]” is justified as the list (Pn)n constitutes (or is equivalentto) a [ρ→ρ]–name for f = ulimn Pn; cf. [Wei01, bottom of p.160]. ⊓⊔

The aforementioned other approach to continuous real hypercomputation arisesfrom allowing the fast convergent sequence (Pn)n ⊆ Q[X ] to be computable in ∅′or ∅′′. The ∅′–computable f : [0, 1] → R have in fact already been characterizedby Ho as Claim a) of the following

Lemma 22. a) To a real function f : [0, 1] → R, there exists a ∅′–computablesequence of polynomials (Pn) satisfying Equation (5) if and only if it holds

[ρ→ρ]′[ρ→ρ]′[ρ→ρ]′: There is a computable sequence (Qm) ⊆ Q[X ] converging uniformly(although not necessarily ‘fast’) to f , that is, with f = ulim

m→∞Qm.

b) For an arbitrary oracle A, the sequence (of discrete degrees and numera-tors/denominators of the coefficients of) (Pn)n ⊆ Q[X ] is A′–computableiff there exists an A–computable sequence (Qn,m)

n,m⊆ Q[X ] such that

∀n ∃M ∀m ≥ M : Pn = Qn,m .

c) To a real function f : [0, 1] → R, there exists a ∅′′–computable sequence ofpolynomials (Pn) satisfying Equation (5) if and only if it holds

[ρ→ρ]′′[ρ→ρ]′′[ρ→ρ]′′: There is a computable sequence (Qm) ⊆ Q[X ] s.t. f = ulimi

ulimj

Q〈i,j〉.

Notice the similarity of Claims a+c) to Fact 3b).

Proof. a) See [Ho99, Theorem 16].

22 Martin Ziegler

b) is a straight-forward extension of Shoenfield’s Limit Lemma [Soa87, LemmaIII.3.3] and its generalization to sequences of rational numbers [ZW01,Lemma 4.1].

c) If ∅′′–computable (Pn)n ⊆ Q[X ] satisfies Equation (5), then by virtue ofthe relativization of [Ho99, Theorem 16] there exists some ∅′–computable(Pn)n ⊆ Q[X ] converging to the same f uniformly on [0, 1]. By Claim a) inturn, Pn = ulimmQn,m for some computable sequence (Qn,m) ⊆ Q[X ].Conversely if f = ulimn Pn with Pn := ulimm Qn,m for a computable (Qn,m),then let Pn := Qn,mn

where

mn := min{

m ∈ N∣

∣ ∀k, ℓ ≥ m : ‖Qn,k −Qn,ℓ‖ ≤ 2−n}

. (6)

This sequence (mn)n is well-defined and yields ‖Pn − Pn‖ ≤ 2−n, so f =ulimn Pn = ulimn Pn. Moreover, the minimum in Equation (6) is taken overa co-r.e. set — r := ‖Qn,k − Qn,ℓ‖ · 2n being ρ–computable by virtue of[Wei01,Corollary 6.2.5] and the complementary condition “r > 1” ρ–r.e.open and hence recursive in ∅′. Similar to Equation (6), this ∅′–computablesequence (Pn)n ⊆ Q[X ] converging uniformly though just ultimately to fcan be turned into a ∅′′–computable, fast convergent one. ⊓⊔

We thus have two hierarchies of hypercomputable continuous real functions:• [ρ→ρ], [ρ→ρ]′, [ρ→ρ]′′, . . .• (ρ → ρ), (ρ′ → ρ′), (ρ′′ → ρ′′), . . .By the Fact 21, their respective ground-levels coincide. Our next result comparestheir respective higher levels. They turn out to lie skewly to each other (Claim c).

Theorem 23. a) Let f : [0, 1] → R be [ρ → ρ]′–computable (in the sense ofLemma 22a). Then, f is (ρ′ → ρ′)–computable.

b) Let f : [0, 1] → R be (ρ′ → ρ′)–computable. Then, f is [ρ→ρ]′′–computable.c) There is a (ρ′ → ρ′)–computable but not [ρ→ρ]′–computable f : [0, 1] → R.

The idea to c) is that every [ρ→ρ]′–computable f : [0, 1] → R has a modulus ofuniform continuity recursive in ∅′; whereas a (ρ′ → ρ′)–computable f , althoughuniformly continuous as well, in general does not.

Before proceeding to the proof, we first provide a tool which turns out tobe useful in the sequel. It is well-known in Recursive Analysis that, althoughequality of real numbers is ρ–undecidable due to the Main Theorem, inequalityis at least semi-decidable. The following lemma generalizes this to ρ′ and to(ρ′ → ρ′<)–computable functions:

Lemma 24. a) Let f : R → R be (ρ → ρ<)–computable. Then the property{

(a, b, c) ∈ Q3∣

∣∃x ∈ [a, b] : f(x) > c}

whether f on [a, b] exceeds c is semi-decidable.b) Let f : R → R be (ρ′ → ρ′<)–computable. Then the property

{

(a, b, c) ∈ Q3∣

∣∃x ∈ [a, b] : f(x) > c}

whether f on [a, b] exceeds c is semi-decidable relative to ∅′.

Real Hypercomputation and Continuity 23

c) Let f : R → R be (ρ′ → ρ′)–computable. Then the property

{

(a, b, c,m) ∈ Q3 × N∣

∣ ∀x ∈ [a, b] : c− 2−m ≤ f(x) ≤ c+ 2−m}

is decidable relative to ∅′′.

Proof. a) is standard; c) follows from b) which is established as follows:By lower semi-continuity of f due to Theorem 11a), if f exceeds c on the compactinterval [a, b], then it does so on some rational x. Feeding, for any such x ∈[a, b] ∩Q, the ρ′–name (x, x, x, x, . . .) for x into the Type-2 Machine computingf reveals the mapping Q ∋ x 7→ f(x) to be (νQ → ρ′<)–computable. With ∅′–oracle, it thus becomes (νQ → ρ<)–computable by virtue of [ZW01, Lemma 4.2].Since {(y, c) : y > c} is (ρ< × νQ)–semi-decidable, the claim follows. ⊓⊔

Proof (Theorem 23).

a) Let (Pn) ⊆ Q[X ] denote a computable sequence converging uniformly (yetnot necessarily fast) to f . Let x ∈ [0, 1] be given as the limit of a sequence(qn) ⊆ Q. Then, pn := Pn(qn) eventually converges to f(x).

b) Let x ∈ [0, 1] be given by (an equivalent to) its ρ–name in form of two rationalsequences (an) and (bn) with {x} =

⋂

n[an, bn]. There exists a rational se-quence (cm) forming a ρ–name for f(x), that is, satisfying c−2−m ≤ f(x) ≤c + 2−m for all m; and by virtue of Lemma 24c), such a sequence can befound with the help of a ∅′′–oracle. This reveals that f is ∅′′–recursive in thesense of [Ho99, Section 4] and thus, similarly to [Ho99, Corollary 17],[ρ→ρ]′′–computable.

c) Let h : N → N denote a ∅′–computable injective total enumeration of somesubset H = h[N] ∈ Σ2 \∆2. Observe that am := 2−h(m) is a ρ′–computablereal sequence converging to 0 with modulus of convergence [Wei01, Def-inition 4.2.2] lacking ∅′–recursivity; compare [Wei01, Exercise 4.2.4c)].Let ϕ : R → R denote some (ρ → ρ)–computable unit pulse, that is, van-ishing outside [0, 1] and having height maxx ϕ(x) = ϕ(12 ) = 1; a piecewiselinear ‘hat ’ function for instance will do fine but we can even choose ϕ as in[PER89, Theorem 1.1.1] to obtain the counter-example

f(x) :=∑

m∈N

am · ϕ(2mx− 1) , (7)

(that is, a non-overlapping superposition of scaled translates of such pulses)to be C∞. By Theorem 15a), x 7→ am ·ϕ(2mx− 1) is (ρ′ → ρ′)–computable;in fact even uniformly in m: Given (qn)n ⊆ Q with x = limn qn, onecan for each M ∈ N obtain a sequence (pk,M )

k⊆ Q with limk pk,M =

∑

m≤MMM am ·ϕ(2mx−1) =: fM . The functions fM converge uniformly (thoughnot effectively) to f because of the disjoint supports of the terms ϕ(2mx−1)in Equation (7). Therefore lim

MpM,M = f(x), thus establishing (ρ′ → ρ′)–

computability of f .Suppose f was [ρ→ρ]′–computable. Then, by virtue of [Ho99, Lemma 15],

24 Martin Ziegler

it has a ∅′–recursive modulus of uniform continuity; cf. [Wei01, Defini-tion 6.2.6.2]. In particular given n ∈ N, one can ∅′–compute m ∈ N suchthat x := 2−m and y := 3

2x satisfy 2−n ≥ |f(x) − f(y)| = |0 − am|contradicting that (am) has no ∅′–recursive modulus of continuity. ⊓⊔

5 Type-2 Nondeterminism

Concerning the two kinds of real hypercomputation considered so far—based onoracle-support and weak real number encodings that is—recall that the accord-ing proofs of Fact 10 and Theorem 11 crucially rely on the underlying TuringMachines to behave deterministically. This raises the question whether nondeter-minism might yield the additional power necessary for evaluating discontinuousreal functions like Heaviside’s.

In the discrete (i.e., Type-1) setting where any computation is required to ter-minate, the finitely many possible choices of a nondeterministic machine can ofcourse be simulated by a deterministic one—however already here subject to theimportant condition that all paths of the nondeterministic computation indeedterminate, cf. [STvE89]. In contrast, a Type-2 computation realizes a trans-formation from/to infinite strings and is therefore a generally non-terminatingprocess. Therefore, nondeterminism here involves an infinite number of guesseswhich turns out cannot be simulated by a deterministic Type-2 machine.

We also point out that nondeterminism has already before been revealed notonly a useful but indeed the most natural concept of computation on Σω. Moreprecisely, Buchi extended Finite Automata from finite to infinite strings andproved that here, as opposed deterministic, nondeterministic ones are closed un-der complement [Tho90] and thus the appropriate model of computation. Since

Chomsky-Level Σ∗ Σω

3: regular Finite Automata Buchi Automata (nondeterministic)

2: context-free

1: context-sensitive

0: unrestricted (Type-1) Turing Machines nondeterministic Type-2 Machines

Fig. 7. Models of Computation in Chomsky’s Hierarchies over finite/infinitestrings

automata and Turing Machines constitute the bottom and top levels, respec-tively, of Chomsky’s Hierarchy of classical languages L ⊆ Σ∗ (Type-1 setting),we suggest that over infinite strings Σω (Type-2 setting) both their respec-tive counterparts, that is Buchi Automata and Type-2 Machines be considerednondeterministically; compare Figure 7.

The concept of nondeterministic computation of a function f :⊆ Σ∗ →Σ∗ (as opposed to a decision problem) is taken from the famous Immerman-Szelepscenyi Theorem in computational complexity; cf. for instance [Pap94,

Real Hypercomputation and Continuity 25

the paragraph preceding Theorem 7.6]: For x ∈ dom(f), some computing pathsof the according machine M may fail by leading to rejecting states, as long as

1) there is an accepting computation of M on x and2) every accepting computation of M on x yields the correct output f(x).

This notion extends straight-forwardly from Type-1 to the Type-2 setting:

Definition 25. Let A and B be sets with respective representations α :⊆ Σω →A and β :⊆ Σω → B. A function f :⊆ A → B is called nondeterministically(α → β)–computable if some nondeterministic one-way Turing Machine M,

– upon input of any α–name σ ∈ Σω for some a ∈ dom(f),– has a computation which outputs a β–name for b = f(a) and– every infinite computation‖ of M on σ outputs a β–name for b = f(a).

This definition is sensible insofar as it leads to closure under composition:

Observation 26. Let f :⊆ A → B be nondeterministically (α → β)–computableand g :⊆ B → C be nondeterministically (β → γ)–computable. Then, g ◦ f :⊆A → C is nondeterministically (α → γ)–computable.

A subtle point in Definition 25, the nondeterministic machine may ‘withdraw’ aguess as long as it does so within finite time.

Example 27 (‘Deciding’ the Arithmetic Hierarchy). Let P ⊆ N be recursive,

A ={

x ∈ N | ∀y1 ∈ N∃z1 ∈ N ∀y2∃z2 . . . ∀yk∃zk : 〈x; y1, z1, . . . , yk, zk〉 ∈ P}

on (or below) level Π2k of Kleene’s Arithmetic Hierarchy. Then the functionχA : N → {0, 1} × { }ω is nondeterministically computable:Observe that x ∈ A iff

∃f1, f2, . . . , fk : N → N ∀y1, y2, . . . , yk ∈ N : 〈x; y1, f(y1), . . . , yk, f(yk)〉 ∈ P

So given x ∈ N, let M+ output “1” and then verify, while continuously spittingout blanks “ ”, that χA(x) = 1 indeed holds. To this end, the machine starts‘guessing’ the values of f = (f1, . . . , fk) restricted to {0, 1, . . . , n} for n = 1, 2, . . .Simultaneously by means of dove-tailing, M+ tries all y ∈ {0, 1, . . . , n}k andaborts in case that the assertion “〈x; y1, f(y1), . . . , yk, f(yk)〉 ∈ P” fails.Now if x ∈ A, then an appropriate f exists, is ultimately ‘found’ by M+, andleads to indefinite execution; whereas if x 6∈ A, then M+ will eventually termi-nate for any guessed f .Since N \ A ∈ Π2k+2, a machine M− can output “0” and then similarly verifyχA(x) = 0. The final machine M, upon input of x ∈ N, nondeterministicallychooses to proceed either like M+ or like M−. Its computation satisfies therequirements of Definition 25. ⊓⊔‖ This condition is slightly stronger than the one required in [Zie05, Definition 14].

26 Martin Ziegler

The power of nondeterministic computation permits conversion forth and backamong representations on the Real Arithmetic Hierarchy from Definition 2:

Theorem 28 (Third Main Theorem of Real Hypercomputation). Foreach d = 0, 1, 2, . . ., the identity R ∋ x 7→ x is nondeterministically (ρ(d+1) →ρ(d))–computable. It is furthermore nondeterministically (ρ → ρb,2)–computable.

Proof. Consider first the case d = 0. Let x ∈ R be given by a sequence (qn) ⊆ Qeventually converging to x. Then, there exists a fast convergent Cauchy sub-sequence (qnk

)k, that is, satisfying

∀k ≥ ℓ : |qnk− qnℓ

| ≤ 2−ℓ−1 (8)

and thus forming a ρ–name for x. To find this subsequence, guess iteratively foreach k ∈ N some nk > nk−1 and check whether it complies with Inequality (8)for the (finitely many) ℓ ≤ k; if it does not, we may abort this computation inaccordance with Definition 25.

For d = 1, let x = limn xn with xn = limm qn,m. Then apply the case d = 0to convert for each n the ρ′–name (qn,m)

mof xn ∈ R into an according ρ–name,

that is, a sequence pn,m satisfying |xn − pn,m| ≤ 2−m. Its diagonal (pn,n)n thenhas |x− pn,n| ≤ |x− xn|+ 2−n → 0 and is thus a ρ′–name for x. Higher levels dcan be treated similarly by induction.

For (ρ → ρb,2)–computability, let x ∈ (0, 2) be given by a fast convergentsequence (qn) ⊆ Q. We guess the leading digit b ∈ {0, 1} for x’s binary expansionb.∗; in case b = 0, check whether x > 1—a ρ–semi decidable property—and if so,abort; similarly in case b = 1, abort if it turns out that x < 1. Otherwise (thatis, proceeding while simultaneously continuing the above semi-decision processvia dove-tailing) replace x by 2(x− b) and repeat guessing the next bit. ⊓⊔

It is also instructive to observe how, in the case of non-unique binary expansion(i.e., for dyadic x), nondeterminism in the above (ρ → ρb,2)–computation gen-erates, in accordance with the third requirement of Definition 25, both possibleexpansions.

Theorem 28 implies that nondeterministic computability of real functionsis largely independent of the representation under consideration — in strikingcontrast to the classical case (Corollary 16) where the effectivity subtleties arisingfrom different encodings had confused already Turing himself [Tur37].

Corollary 29. a) With respect to nondeterministic reduction “�n” , it holdsρb,2 ≡n ρ ≡n ρ< ≡n ρ′ ≡n ρ′< ≡n ρ′′ ≡n . . ..

b) The entire Real Arithmetic Hierarchy of Weihrauch and Zheng is non-deterministically computable.

Proof. a) follows from Lemma 5 and Theorem 28.b) Let x ∈ ∆d+1R for some d ∈ N. Then, x ∈ R is ρ(d)–computable by Defini-

tion 2; hence nondeterministically also ρb,2–computable by a). Alternativelycombine Example 27 with Fact 3a). ⊓⊔

Real Hypercomputation and Continuity 27

In particular, this kind of hypercomputation allows for nondeterministic (ρ →ρ)–evaluation of Heaviside’s function by appending to the (ρ → ρ<)–computationin Example 9 a conversion from ρ< � ρ′ back to ρ. Section 5.1 establishes manymore real functions, both continuous and discontinuous ones, to be nondeter-ministically computable, too.

5.1 Nondeterministic and Analytic Computation

We now show that Type-2 nondeterminism includes the algebraic so called BCSS-model of real number computation due to Blum, Cucker, Shub, and Smale[BSS89,BCSS98] employed for instance in Computational Geometry [PS85, Sec-tion 1.4]. As a matter of fact, nondeterministic real hypercomputation evencovers all quasi-strongly δ–Q–analytic functions f :⊆ Rd → R in the sense ofChadzelek and Hotz [CH99, Definition 5]. The latter can be considereda synthesis of the Type-2 (i.e., infinite approximate) and the BCSS (i.e., finiteexact) model of real number computation. Its computational power admits anelegant characterization (see Lemma 31b+c) in terms of the following

Definition 30. A ρH–name for x ∈ R is some (qn)n ⊆ Q such that

∃N ∀n ≥ N : |qn − x| ≤ 2−n . (9)

The encoding sequence of rational approximations must thus converge fast withthe exception of some initial segment of finite yet unknown length. It correspondsto ρ–computation by an Inductive Turing Machine in the sense of [Bur04] whichis roughly speaking a Type-2 Machine but whose output tape(s) need not be one-way [Wei01, Section 2.1] provided that the contents of every cell ultimatelystabilizes.

Lemma 31. a) It holds ρ ¬ ρH ¬ ρ′.b) A function f :⊆ RN → R is (ρN → ρH)–computable iff it is computable

by a quasi-strongly δ–Q–analytic machine.c) (ρ → ρH)–computability is equivalent to (ρH → ρH)–computability.d) The class of (ρH → ρH)–computable functions is closed under composition.The above claims relativize.

Proof. a) is immediate.b) Observe that the robustness of the program π required in [CH99, top of

p.157] amounts to the argument x ∈ R of f being accessible by rationalapproximations qn ∈ Q of error |qn − x| ≤ 2−n, that is, in terms of a ρ–name. The output y = f(x) on the other hand proceeds by way of twosequences (pm)

m, (ǫm)

m⊆ Q such that ǫm → 0 and |pm−y| ≤ ǫm holds for all

sufficiently largem. By effectively proceeding to an appropriate subsequence,we can w.l.o.g. suppose ǫm = 2−m, hence (pm) is ρH–name of y.

c) By a), every (ρH → ρH)–computable function is (ρ → ρH)–computable, too.For the converse implication, take the Type-2 Machine M converting ρ–names for x ∈ R to ρH–names for y = f(x). Let (qn) satisfy Equation (9) forsome unknown N ∈ N.

28 Martin Ziegler

Now simulate M on (qn)n≥0, implicitly supposing that it is a valid ρ–name,

i.e., that N = 0. Simultaneously check consistency of Condition (9), thatis, verify |qn − qk| ≤ 2−n+1∀k ≥ n ≥ N . If (or, rather, when) the latterfails for some (k0, n0), M has output only finitely (say M0 ∈ N) manypm ∈ Q. In that case, restart M on (qn)n≥1

presuming N = 1 while, again,checking this presumption consistent with (9); but this time throw away thefirst M0 elements of the sequence printed by M. Continue analogously forN = 2, 3, . . ..We claim that this yields output of a ρH–name for y. Since (qn) is a valid ρH–name, a feasible N will eventually be found. Before that happens, the severalpartial runs of M have produced only finitely (say M ∈ N) many rationalnumbers pm; and after that, the final simulation generates by presumptiona valid ρH–name for y. Out of this sequence (pm)

m, the first M entries

may have been exchanged by outputs of previous simulation trials; howeveraccording to Definition 30, the representation ρH is immune against suchfinite modifications.

d) Quasi-strongly δ–Q–analytic functions are closed under composition accord-ing to [CH99, Lemma 2]; now apply b+c). ⊓⊔

A BCSS (or, equivalently, an R–) machine M is permitted to store a finitenumber of arbitrary real constants r1, . . . , rk [CH99, Instruction 1(b) in Table 1on p.154] and use it for instance to solve the Halting or any other fixed discreteproblem [BSS89, Example 6]. Slightly correcting [CH99, Theorem 3], M’ssimulation by a rational machine thus requires knowledge of r := (r1, . . . , rk) ∈Rk; e.g. by virtue of oracle access to (O := {bin(n) : σn = 1} ⊆ {0, 1}∗ asnatural encoding of a ρkb,2–name σ ∈ {0, 1}ω of) r—compare [BV99] for the caseof simulating ρ–semi decidability.

Proposition 32. a) A function f :⊆ R → R computable by a BCSS–machinewith constants r ∈ Rk is also (ρH → ρH)–computable relative to r.

b) Every (ρ → ρ)–computable function f :⊆ R → R is also (ρH → ρH)–computable.

c) Let f :⊆ R → R be (ρH → ρH)–computable relative to some oracle O ⊆ Σ∗

in (Kleene’s) Arithmetic Hierarchy. Then f is nondeterministically Type-2computable.

Proof. a) See (the proof of) [CH99, Theorem 3].b) Combine Lemma 31a+c).c) The nondeterministic simulation can answer queries to O due to Example 27.

As ρ ≡n ρH ≡n ρ′ by Corollary 29a) and Lemma 31a), the claim follows. ⊓⊔

Let us illustrate Proposition 32a) with the following

Example 33. Heaviside’s Function h : R → {0, 1} is trivially BCSS–computable.It is also (ρH → ρH)–computable by means of conservative branching: Givenx ∈ R by virtue of (qn) ⊆ Q with (9) and unknown N ∈ N, let pn := 0 ifqn ≤ 2−n and pn := 1 otherwise.Indeed if x ≤ 0 then, for all n ≥ N , qn ≤ 2−n and thus pn = 0 = f(x). If on the

Real Hypercomputation and Continuity 29

other hand x > 0, x > 2−M for some M ∈ N; then, for all n ≥ max{M + 1, N},qn > 2−n so pn = 1 = f(x). ⊓⊔

Of course the class of nondeterministic Type-2 Machines (and thus also thatof the nondeterministically computable real functions) is still only countablyinfinite: most (even constant) functions f : R → R actually remain infeasible tothis kind of real hypercomputation.

6 Conclusion

Recursive Analysis is often criticized for being unable, due to its Main Theo-rem, to non-trivially treat discontinuous functions. Although one can in Type-2Theory devise sensible computability notions for, say, generalized (and in par-ticular discontinuous) functions as for instance in [ZW03], evaluation x 7→ f(x)of an L2 function or a distribution f at a point x ∈ R does not make sensehere already mathematically. Regarding the Main Theorem’s connection to theChurch-Turing Hypothesis indicated in the introduction, the present work hasinvestigated whether and which models of hypercomputation allows for liftingthat restriction.

A first idea, relativized computation on oracle Turing Machines, was ruledout right away. A second idea, computation based on weakened encodings ofreal numbers, renders evaluation x 7→ h(x) of Heaviside’s function—althoughdiscontinuous—for instance (ρ → ρ<)–computable. The drawback of this notionof real hypercomputation: it lacks closure under composition.

Example 34. Let f : R → R, f(0) := 0 and f(x) := 1 for x 6= 0. Letg(x) := −x. Then both f and g are (ρ → ρ<)–computable but their compositiong ◦ f : 0 7→ 0, 0 6= x 7→ −1 lacks lower semi–continuity.

Requiring both argument x and value y = f(x) to be encoded in the same way—say, ρ, ρ′, or ρ′′—asserts closure under both composition and negation f 7→ −f ;and the prerequisites of the Main Theorem applies only to the case (ρ → ρ).Surprisingly, (ρ′ → ρ′)–computability and (ρ′′ → ρ′′)–computability still requirecontinuity! These results extend to (ρ(d) → ρ(d))–computability for arbitraryd, although already the step from d = 1 to 2 made proofs significantly moreinvolved.

These claims immediately relativize, that is, even a mixture of oracle supportand weak real number encodings does not allow for hypercomputational evalua-tion of discontinuous functions. This is due to the purely information-theoreticnature of the arguments employed, specifically: the deterministic behavior of theTuring Machines under consideration.

So we have finally looked at nondeterminism as a further way of enhancingthe underlying machine model beyond Turing’s barrier. Over the Type-2 settingof infinite strings Σω, this parallels Buchi’s well-established generalization offinite automata to so-called ω–regular languages. While the practical realizabilityof Type-2 nondeterminism is admittedly even more questionable than that of

30 Martin Ziegler

classical NP-machines, it does yield an elegant notion of hypercomputationwith nice closure properties and invariant under various encodings.

A precise characterization of the class of nondeterministically computablereal functions will be subject of future work.

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